Infrared radiation in potential scattering

Some time ago Bloch and Nordsieck, working in a model in which the spectrum of the radiation field is cut off beyond some low frequency ${\omega }_1$, showed that the transition probability for scattering, summed over final photon states, is approximately that which would be obtained if the interaction of the projectile with the radiation field were neglected entirely. Here, within the context of a nonrelativistic treatment of the scattering process, the sum rule is generalized through the inclusion of corrections of first order in ${\omega }_1$. These corrections can be interpreted in terms of a simple classical picture. In the course of the derivation, a lowfrequency approximation for the transition amplitude is obtained which contains as special cases the perturbative result of Low for single-photon bremsstrahlung and the more recently derived approximation for scattering in a low-frequency laser field.